a fraction becomes 9/11 if 2 is added to bith numerator and denominato

1.	a fraction becomes 9/11 if 2 is added to bith numerator and denominator . if 3 is added to both numerator and denominator it becomes 5/6 find the fraction
Answer» Given : A fraction BECOMES 9/11 if 2 is ADDED to both numerator and denominator. If 3 is added to numerator and denominator both fraction becomes 5/6. To Find : The fraction. Solution : Let the numerator of the fraction be x. Let the denominator of the fraction be y. Fraction = $\bold{\dfrac{x}{y}}$ Case 1 : When 2 is added to the numerator and denominator of the fraction, it becomes 9/11. Numerator = (x+2) Denominator = (y+2) EQUATION : $\implies$ $\sf{\dfrac{x+2}{y+2}\:=\:\dfrac{9}{11}}$ $\implies$ $\sf{11(x+2)=9(y+2)}$ $\implies$ $\sf{11x+22=9y+18}$ $\implies$ $\sf{11x-9y=18-22}$ $\implies$ $\sf{11x-9y=-4}$ $\implies$ $\sf{11x=-4+9y}$ $\large{\sf{x=\dfrac{-4+9y}{11}\:\:\:(1)}}$ Case 2 : If 3 is added to the numerator and denominator, the fraction becomes 5/6. Numerator = (x+3) Denominator = (y+3) Equation : $\implies$ $\sf{\dfrac{x+3}{y+3}\:=\:\dfrac{5}{6}}$ $\implies$ $\sf{6(x+3)=5(y+3)}$ $\implies$ $\sf{6x+18=5y+15}$ $\implies$ $\sf{6x-5y=15-18}$ $\implies$ $\sf{6x-5y=-3}$ $\implies$ $\sf{6\:\Big(\dfrac{-4+9y}{11}\Big)-5y=-3}$ $\bold{\Big[From\:equation\:(1)\:x\:=\:\dfrac{-4+9y}{11}\Big]}$ $\implies$ $\sf{\dfrac{-24+54y}{11}-5y\:=\:-3}$ $\implies$ $\sf{\dfrac{-24+54y-55y}{11}\:=\:-3}$ $\implies$ $\sf{-24+54y-55y=11\:\times\:-3}$ $\implies$ $\sf{-24+54y-55y=-33}$ $\implies$ $\sf{54y-55y=-33+24}$ $\implies$ $\sf{-y=-9}$ $\implies$ $\sf{y=9}$ SUBSTITUTE, y = 9 in equation (1), $\implies$ $\sf{x=\dfrac{-4+9y}{11}}$ $\implies$ $\sf{x=\dfrac{-4+9(9)}{11}}$ $\implies$ $\sf{x=\dfrac{-4+81}{11}}$ $\implies$ $\sf{x=\dfrac{77}{11}}$ $\implies$ $\sf{x=7}$ $\large{\boxed{\bold{Numerator\:of\:the\:fraction\:=\:x\:=\:7}}}$ $\large{\boxed{\bold{Denominator\:of\:the\:fraction\:=\:y\:=\:9}}}$ $\large{\boxed{\bold{\red{Fraction\:=\:\dfrac{x}{y}\:=\:\dfrac{7}{9}}}}}$